1 / 22

SEVENTH EDITION and EXPANDED SEVENTH EDITION

SEVENTH EDITION and EXPANDED SEVENTH EDITION. Chapter 10. Mathematical Systems. 10.1. Groups. Definitions. A mathematical system consists of a set of elements and at least one binary operation.

lorit
Download Presentation

SEVENTH EDITION and EXPANDED SEVENTH EDITION

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. SEVENTH EDITION and EXPANDED SEVENTH EDITION

  2. Chapter 10 Mathematical Systems

  3. 10.1 Groups

  4. Definitions • A mathematical system consists of a set of elements and at least one binary operation. • A binary operation is an operation, or rule, that can be performed on two and only two elements of a set.

  5. For elements a, b, and c Addition Multiplication Commutative property a + b = b + a ab = ba Associate property (a + b) + c = a + (b + c) (ab)c = a(bc) Properties

  6. Closure • If a binary operation is performed on any two elements of a set and the result is an element of the set, then that set is closed (or has closure) under the given binary operation.

  7. Identity Element • An identity element is an element in a set such that when a binary operation is performed on it and any given element in the set, the result is the given element. • Additive identity element is 0. • Multiplicative identity element is 1.

  8. Inverses • When a binary operation is performed on two elements in a set and the result is the identity element for the binary operation, each element is said to be the inverse of the other.

  9. Properties of a Group • Any mathematical system that meets the following four requirements is called a group. • The set of elements is closed under the given operation. • An identity element exists for the set under the given operation. • Every element in the set has an inverse under the given operation. • The set of elements is associative under the given operation.

  10. Commutative Group • A group that satisfies the commutative property is called a commutative group or (abelian group).

  11. Properties of a Commutative Group • A mathematical system is a commutative group if all five conditions hold. • The set of elements is closed under the given operation. • An identity element exists for the set under the given operation. • Every element in the set has an inverse under the given operation. • The set of elements is associative under the given operation. • The set of elements is commutative under the given conditions.

  12. 10.2 Finite Mathematical Systems

  13. Definition • A finite mathematical system is one whose set contains a finite number of elements. • Example: Determine whether the clock arithmetic system under the operation of addition is a commutative group.

  14. Definition continued • Closure: The set of elements in clock arithmetic is closed under the operation of addition. • Identity: There is an additive identity element, namely 12. • Inverse elements: Each element in the set has an inverse. • Associative property: The system is associative under the operation of addition. • Commutative property: The commutative property of addition is true for clock arithmetic. • The system satisfies the five properties required for a mathematical system. Thus, clock arithmetic under the operation of addition is a commutative or abelian group.

  15. 10.3 Modular Arithmetic

  16. Definition • A modulo m system consists of m elements, 0 through m 1, and a binary operation. • a is congruent to b modulo m, written a  b(mod m), if a and b have the same remainder when divided by m.

  17. Example • Determine which number from 0 to 7, the following numbers are congruent to in modulo 8. • a) 66 b) 72 c) 109

  18.  remainder Solution: a) 66 • To determine the value 66 is congruent to in mod 8, divide 66 by 8 and find the remainder. Thus, 66  2 (mod 8)

  19. b) 72 72  ? (mod 8) 72  8 = 9, remainder 0 72  0 (mod 8) c) 109 109  ? (mod 8) 109  8 = 13, remainder 5 109  5 (mod 8) Solutions continued

  20. Example • Evaluate each in mod 6. • a) 4 + 2 b) 3  2 c) 2(4) • a) 4 + 2  ? (mod 6) 6  ? (mod 6) 6  6 = 1, remainder 0. Therefore, 4 + 2  0 (mod 6)

  21. b) 3  2 3  2  ? (mod 6) 1  ? (mod 6) 1  1 (mod 6) c) 4(2) 4(2)  ? (mod 6) 8  ? (mod 6) 8  6 = 1, remainder 2. Therefore, 4(2)  2 (mod 6) Example continued

  22. a) 4 • ?  3(mod 5) One method is to replace the ? mark with the numbers. 4 • 0  0(mod 5) 4 • 1  4(mod 5) 4 • 2  3(mod 5) 4 • 3  2(mod 5) 4 • 4  1(mod 5) Therefore, ? = 2 since 4 • 2  3(mod 5). b) 3 • ?  0(mod 5) 3 • 0  0(mod 5) 3 • 1  3(mod 5) 3 • 2  0(mod 5) 3 • 3  3(mod 5) 3 • 4  0(mod 5) 3 • 5  3(mod 5) Therefore, the 0, 2 and 4 result in true statements. Find all replacements for the question mark that make the statements true.

More Related